Obvious natural morphisms of sheaves are unique

Ryan Cohen Reich

We prove that a large class of natural transformations (consisting
roughly of those constructed via composition from the ``functorial''
or ``base change'' transformations) between two functors of the form
$... f^* g_* ...$ actually has only one element, and thus that
any diagram of such maps necessarily commutes. We identify the
precise axioms defining what we call a ``geofibered category'' that
ensure that such a coherence theorem exists. Our results apply to
all the usual sheaf-theoretic contexts of algebraic geometry. The
analogous result that would include any other of the six functors
remains unknown.